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Properties

Label 2-1950-13.10-c1-0-41
Degree $2$
Conductor $1950$
Sign $-0.862 + 0.505i$
Analytic cond. $15.5708$
Root an. cond. $3.94598$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)2-s + (0.5 + 0.866i)3-s + (0.499 − 0.866i)4-s + (0.866 + 0.499i)6-s + (−1.31 − 0.758i)7-s − 0.999i·8-s + (−0.499 + 0.866i)9-s + (−4.45 + 2.57i)11-s + 0.999·12-s + (−3.08 + 1.86i)13-s − 1.51·14-s + (−0.5 − 0.866i)16-s + (2.39 − 4.15i)17-s + 0.999i·18-s + (−1.39 − 0.807i)19-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.288 + 0.499i)3-s + (0.249 − 0.433i)4-s + (0.353 + 0.204i)6-s + (−0.496 − 0.286i)7-s − 0.353i·8-s + (−0.166 + 0.288i)9-s + (−1.34 + 0.775i)11-s + 0.288·12-s + (−0.856 + 0.516i)13-s − 0.405·14-s + (−0.125 − 0.216i)16-s + (0.581 − 1.00i)17-s + 0.235i·18-s + (−0.320 − 0.185i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.862 + 0.505i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.862 + 0.505i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-0.862 + 0.505i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (751, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ -0.862 + 0.505i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8404622184\)
\(L(\frac12)\) \(\approx\) \(0.8404622184\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 \)
13 \( 1 + (3.08 - 1.86i)T \)
good7 \( 1 + (1.31 + 0.758i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (4.45 - 2.57i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.39 + 4.15i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.39 + 0.807i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.72 + 4.71i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.93 + 8.55i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 9.88iT - 31T^{2} \)
37 \( 1 + (-7.56 + 4.36i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.10 + 0.637i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.18 - 2.04i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 6.84iT - 47T^{2} \)
53 \( 1 - 0.881T + 53T^{2} \)
59 \( 1 + (8.04 + 4.64i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.24 - 10.8i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (9.98 - 5.76i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-13.7 - 7.91i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 4.39iT - 73T^{2} \)
79 \( 1 + 11.0T + 79T^{2} \)
83 \( 1 - 0.979iT - 83T^{2} \)
89 \( 1 + (3.54 - 2.04i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.95 - 2.85i)T + (48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.200352614018372828220302749083, −7.77082457325216665673219111524, −7.51237208520509376556677197115, −6.30946130279899772561388603833, −5.47347847781662420301365006840, −4.53880516825540591989781098123, −4.08682307079628202605546537291, −2.67299270955487886097244252911, −2.34732640898367017417715636011, −0.21100331669408798440934115006, 1.74823722923232023759000810007, 3.03740496139955930053433784658, 3.37527117705658961637178603741, 4.86013981509708791991590655114, 5.60423804815548750772251093252, 6.19616436068744884814688078865, 7.23051778200235849402131531385, 7.86381605217255889484767617788, 8.437697546785590319178864028232, 9.396879621952205043712944382878

Graph of the $Z$-function along the critical line